Our aim in this paper is to apply the approach of [1]} to weighted Rota-Baxter Lie algebras. More precisely, we first consider representations of weighted Rota-Baxter Lie algebras and provide various constructions of representations. Then we define the cohomology of a weighted Rota-Baxter Lie algebra with coefficients in a representation. This cohomology is obtained as a byproduct of the standard Chevalley-Eilenberg cohomology of the underlying Lie algebra and the cohomology of the underlying weighted Rota-Baxter operator. When the weight is zero, our cohomology coincides with the one introduced in [2]}. We interpret our second cohomology group as the isomorphism classes of abelian extensions of weighted Rota-Baxter Lie algebras. Then we consider deformations of weighted Rota-Baxter Lie algebras in which we simultaneously deform underlying Lie algebras and weighted Rota-Baxter operators. The infinitesimals of such deformations are 2-cocycles in the cohomology and equivalent deformations produce cohomologous 2-cocycles. Hence they correspond to the same element in the second cohomology group. We also find a sufficient condition for the rigidity of a weighted Rota-Baxter Lie algebra.
